TE and TM modes

A transverse-electric (TE) mode is defined by E_z=0 with H_z nonzero. A transverse-magnetic (TM) mode is defined by H_z=0 with E_z nonzero. The words 'transverse' refer to ONE field relative to the guide axis, not to both fields.

A TEM mode would require E_z=H_z=0 and was excluded in chapter 01 for a simply connected hollow single-conductor guide. Therefore every nontrivial hollow-guide mode belongs to a TE or TM family, apart from possible hybrid modes introduced by more complicated material/geometric structures.

These are modal classifications, not polarization labels in the plane-wave sense. A TE guide mode still possesses a longitudinal magnetic field; a TM guide mode still possesses a longitudinal electric field.

Use exp(+jωt−jβz), k²=ω²με and k_c²=k²−β². Splitting the two curl equations into transverse and longitudinal parts gives a linear system for E_t and H_t. Solving it produces the general reconstruction formulas.

The first term in each expression comes from the gradient of the same field's longitudinal component; the cross-product term couples the opposite longitudinal component. These formulas are valid for k_c≠0 in a homogeneous uniform guide.

Once a longitudinal eigenfunction and its amplitude are chosen, the transverse vector fields are no longer independent unknowns. This is why solving one scalar eigenproblem is enough to generate a complete TE or TM mode.

Set E_z=0 in the general formulas. The longitudinal magnetic eigenfunction H_z obeys the Neumann wall condition from chapter 01. Its transverse gradient generates H_t directly, while a ninety-degree transverse rotation by ẑ× converts that gradient into E_t.

The TE field pair is therefore geometrically locked: E_t is orthogonal in the transverse plane to ∇_tH_z, while H_t is parallel to ∇_tH_z up to complex coefficients.

Multiplying H_z by any complex constant multiplies the entire TE mode by that same constant. Boundary/eigenvalue analysis fixes shape and propagation constant, not absolute excitation amplitude.

Set H_z=0. The longitudinal electric eigenfunction E_z obeys homogeneous Dirichlet data on the PEC boundary. Its transverse gradient generates E_t, while the rotated gradient generates H_t.

For TM modes, E_t is parallel to ∇_tE_z and H_t is orthogonal to it in the transverse plane, again up to the complex coefficients fixed by Maxwell.

The same arbitrary normalization freedom remains: the scalar eigenproblem fixes the spatial eigenfunction only up to a multiplicative complex amplitude.

For a single propagating mode, define modal wave impedance as the transverse electric-to-magnetic field ratio with the local vector orientation understood. The reconstruction coefficients give Z_TE=ωμ/β and Z_TM=β/(ωε).

Using η=√(μ/ε) and β/k, the same relations become Z_TE=η(k/β) and Z_TM=η(β/k). Their product is η². Unlike the TEM plane-wave impedance η, hollow-guide modal impedance depends on the axial propagation constant.

This quantity is a FIELD impedance of one mode. It is not the characteristic impedance Z₀ of RF-00's two-conductor voltage/current line unless a specific voltage/current normalization has been defined for the guide.

Consider two longitudinal scalar eigenfunctions ψ_m and ψ_n satisfying the same Dirichlet or Neumann boundary type with distinct eigenvalues k_cm² and k_cn². Multiply one Helmholtz equation by the complex conjugate of the other, subtract the conjugate equation and integrate over A.

Green's identity converts the difference into a boundary term. The common homogeneous Dirichlet or Neumann condition makes that boundary term vanish, leaving (k_cm²−k_cn²)∫_A ψ_mψ_n* dA=0.

Therefore eigenfunctions with distinct eigenvalues are orthogonal under the cross-sectional inner product. Degenerate modes sharing the same eigenvalue are not automatically selected uniquely; an orthogonal basis can be chosen inside that degenerate eigenspace.

An eigenfunction can be normalized by peak field, by L² norm of its generating scalar, by stored energy or by carried power. These conventions describe the SAME physical mode shape but assign different numerical amplitudes.

For propagating lossless modes, a useful engineering normalization sets the time-average axial power P=(1/2)Re∫_A(E_t×H_t*)·ẑ dA to a chosen value such as one watt. Orthogonal propagating modes then admit independent modal amplitudes under the usual PEC homogeneous-guide assumptions.

Do not compare modal amplitudes from two sources unless their normalization convention is known. A field pattern normalized to max|E|=1 is not numerically interchangeable with a one-watt power-normalized mode.

Suppose one eigenmode is operated at a frequency/material point for which β/k=√3/2. This ratio could arise from many cross-sectional eigenvalues; no rectangular-guide formula is needed.

Then Z_TE=(2/√3)η≈1.155η while Z_TM=(√3/2)η≈0.866η. Their product is exactly η². In vacuum, using η≈376.73 Ω gives approximately 435 Ω for TE and 326 Ω for TM.

As β/k changes, TE and TM impedances move in opposite directions. Chapter 03 will connect that behavior to cutoff and dispersion explicitly.

1. State precisely what TE, TM and TEM mean relative to the guide axis.
2. Starting from the general reconstruction formulas, recover the TE field relations by setting E_z=0.
3. Recover the TM field relations by setting H_z=0.
4. For a TE generating function H_z=H₀cos(πx/a), identify the directions of ∇_tH_z, E_t and H_t away from its nodes.
5. Derive Z_TE=ωμ/β from the TE reconstruction coefficients.
6. Derive Z_TM=β/(ωε) and prove Z_TEZ_TM=η².
7. Why is modal wave impedance not automatically the same as a two-conductor characteristic impedance?
8. Use Green's identity to prove scalar eigenfunction orthogonality for distinct eigenvalues under homogeneous Dirichlet or Neumann conditions.
9. Explain what changes in the orthogonality argument when two independent modes are degenerate.
10. For β/k=0.8 in a medium with η=250 Ω, compute Z_TE and Z_TM and verify their product.

Chapter 03 will turn β²=k²−k_c² into cutoff and dispersion physics: propagating versus evanescent regimes, guide wavelength, phase/group velocities, TE/TM impedance behavior near cutoff and the limits of lossless PEC dispersion formulas.